Wigner distribution ($W[u,v](\mathbf{z},\mathbf{p})$)

The Wigner distribution of two fields is the Fourier transform of a field's two-point product in its separation variable:Its inverse reconstructs $u(\mathbf{z})\overline{v({\mathbf{z}}^{\prime })}$. For vector fields, specify whether the product is an outer product or a scalar contraction; the latter can encode an average over receiver channels.

For $X_1=x$, $X_2=y$ with $[X_1,X_2]=i/2$, normalize the phase-space function by $\int W(x,y)dxdy=1$. A coherent state of amplitude $\alpha$ has $W=2e^{-2|x+iy-\alpha {|}^2}/\pi$. Quadrature marginals are physical probability distributions, but the full function can be negative. Cross terms of an optical cat state give interference fringes, even when low-order moments approach those of a mixture.

For $i{\Psi }_t=-\Delta \Psi /2$, differentiate the two-point product $\Psi (x+y/2)\overline{\Psi }(x-y/2)$. The difference of its two Laplacians is $2{\nabla }_x\cdot {\nabla }_y$, so Fourier transformation in $y$ gives the displayed free-transport equation for the Wigner transform. Its solution is $w(x,\xi ,t)=w(x-t\xi ,\xi ,0)$.

For receiver-dependent Green functions $g_{\mathbf{\zeta }}(\mathbf{z})$ and aperture weight $a(\mathbf{\zeta })$, define ${\mathcal{W}}_A(\mathbf{Z},\mathbf{p})=\int a(\mathbf{\zeta })W[g_{\mathbf{\zeta }},g_{\mathbf{\zeta }}](\mathbf{Z},\mathbf{p})d\mathbf{\zeta }$. Its inverse separation Fourier transform is the kernel of the time reversal operator on the source plane.